Assignment1
Final steps.
- Remove severe outliers in changes and make a revision on univariate outliers [Eliya] Done
- Choose a multivariate outlier removal [Eliya] Done
- Data validation [Achraf]
- Exploratory analisis with some plots and catdesc and condes [Eliya] almost done
- Take a look to transformations [Achraf]
- Weight the age variable to obtain a balanced model [Achraf]
- Make the model validations arguments of why we choose the final model and what we think that could be improved but is not in our hands + Graphs of all Effects [Both]
- Add more reasoning to all steps. [Both]
DATA PREPARATION
• Removing duplicate or irrelevant observations
[Eliya] DONE
• Fix structural errors (usually coding errors, trailing blanks
in labels, lower/upper case consistency, etc.).
[Eliya] DONE
• Check data types. Dates should be coded as such and factors
should have level names (if possible, levels have to be set and clarify
the variable they belong to). This point is sometimes included under
data transformation process. New derived variables are to be produced
sometimes scaling and/or normalization (range/shape changes to numeric
variables) or category regrouping for factors (nominal/ordinal).
[Eliya] DONE
• Filter unwanted outliers. Univariate and multivariate outliers have to be highlighted.Remove register/erase values and set NA for univariate outiers. [Eliya] –I HAVE PATIALLY DID IT, STILL NEED TO UNDERSTAND SOMETHING THERE
• Handle missing data: figure out why the data is missing. Data
imputation is to be considered when the aim is modelling (imputation has
to be validated). [Achraf]
• Data validation is mixed of ‘common sense and sector knowledge’: Does the data make sense? Does the data follow the appropriate rules for its field? Does it prove or disprove the working theory, or bring any insight to light? Can you find trends in the data to help you form a new theory? If not, is that because of a data quality issue? [Achraf]
TASKS
Create factors for qualitative variables. [Eliya]DONEDetermine if the response variable (charges) has an acceptably normal distribution[Achraf]Address tests to discard serial correlation. [Eliya] DONE
Detect univariant and multivariant outliers, errors and missing values (if any) and apply animputation technique if needed.[Achraf]Preliminary exploratory analysis to describe the relationships observed has to be undertaken. [Eliya]
If you can improve linear relations or limit the effect of influential data, you must consider the suitable transformations for variables. [Achraf]
Apart from the original factor variables, you can consider other categorical variables that can be defined from categorized numeric variables. [Eliya]
You must take into account possible interactions between categorical and numerical variables. [Eliya]
When building the model, you should study the presence of multicollinearity and try to reduce their impact on the model for easier interpretation. [Achraf]
You should build the model using a technique for selecting variables (removing no significant predictors and/or stepwise selection of the best models). [Achraf]
The validation of the model has to be done with graphs and / or suitable tests to verify model assumptions. [Achraf]
You must include the study of unusual and / or influential data. [Achraf]
The resulting model should be interpreted in terms of the relationships of selected predictors and its effect on the response variable. [Eliya]
ASSIGNMENT
library(GGally)## Loading required package: ggplot2## Registered S3 method overwritten by 'GGally':
## method from
## +.gg ggplot2#install.packages("data.table")
library(data.table)
library(car)## Loading required package: carDatalibrary(rpart)
library(chemometrics)
#install.packages("mvoutlier")
library(mvoutlier)## Loading required package: sgeostatlibrary(sgeostat)
library(lmtest)## Loading required package: zoo##
## Attaching package: 'zoo'## The following objects are masked from 'package:base':
##
## as.Date, as.Date.numericPreparing the data in the environment
# Clear plots
if(!is.null(dev.list())) dev.off()## null device
## 1# Clean workspace
rm(list=ls())
#load data
df <- read.csv("insurance.csv")Data cleaning
Data format
is.null(df) #no nulls in the data## [1] FALSEreplace(df,which(df %like% " "), '') #close all blank spaces## age sex bmi children smoker region charges
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## 1338which(df=="") #no blanks found in the data## integer(0)#check for distinct values and whether there are differences in them
unique(df$sex) #expecting 2 values## [1] "female" "male"unique(df$smoker) #expecting 2 values## [1] "yes" "no"unique(df$region) #expecting 4 values## [1] "southwest" "southeast" "northwest" "northeast"#we can see that data is consistent for categorical variables
df$f.sex <- factor(df$sex,labels = c("female","male"));
df$f.smoker <- factor(df$smoker,labels = c("no","yes"))
df$f.region <- factor(df$region,labels = c("northeast","northwest","southeast","southwest"))
summary(df) #from the summary we can see the factor values, it seems that sex and region are distributed equally and not much smokers compare to the non smokers.## age sex bmi children
## Min. :18.00 Length:1338 Min. :15.96 Min. :0.000
## 1st Qu.:27.00 Class :character 1st Qu.:26.30 1st Qu.:0.000
## Median :39.00 Mode :character Median :30.40 Median :1.000
## Mean :39.21 Mean :30.66 Mean :1.095
## 3rd Qu.:51.00 3rd Qu.:34.69 3rd Qu.:2.000
## Max. :64.00 Max. :53.13 Max. :5.000
## smoker region charges f.sex f.smoker
## Length:1338 Length:1338 Min. : 1122 female:662 no :1064
## Class :character Class :character 1st Qu.: 4740 male :676 yes: 274
## Mode :character Mode :character Median : 9382
## Mean :13270
## 3rd Qu.:16640
## Max. :63770
## f.region
## northeast:324
## northwest:325
## southeast:364
## southwest:325
##
## dim(df)## [1] 1338 10unique(df)## age sex bmi children smoker region charges f.sex f.smoker
## 1 19 female 27.900 0 yes southwest 16884.924 female yes
## 2 18 male 33.770 1 no southeast 1725.552 male no
## 3 28 male 33.000 3 no southeast 4449.462 male no
## 4 33 male 22.705 0 no northwest 21984.471 male no
## 5 32 male 28.880 0 no northwest 3866.855 male no
## 6 31 female 25.740 0 no southeast 3756.622 female no
## 7 46 female 33.440 1 no southeast 8240.590 female no
## 8 37 female 27.740 3 no northwest 7281.506 female no
## 9 37 male 29.830 2 no northeast 6406.411 male no
## 10 60 female 25.840 0 no northwest 28923.137 female no
## 11 25 male 26.220 0 no northeast 2721.321 male no
## 12 62 female 26.290 0 yes southeast 27808.725 female yes
## 13 23 male 34.400 0 no southwest 1826.843 male no
## 14 56 female 39.820 0 no southeast 11090.718 female no
## 15 27 male 42.130 0 yes southeast 39611.758 male yes
## 16 19 male 24.600 1 no southwest 1837.237 male no
## 17 52 female 30.780 1 no northeast 10797.336 female no
## 18 23 male 23.845 0 no northeast 2395.172 male no
## 19 56 male 40.300 0 no southwest 10602.385 male no
## 20 30 male 35.300 0 yes southwest 36837.467 male yes
## 21 60 female 36.005 0 no northeast 13228.847 female no
## 22 30 female 32.400 1 no southwest 4149.736 female no
## 23 18 male 34.100 0 no southeast 1137.011 male no
## 24 34 female 31.920 1 yes northeast 37701.877 female yes
## 25 37 male 28.025 2 no northwest 6203.902 male no
## 26 59 female 27.720 3 no southeast 14001.134 female no
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## 1336 18 female 36.850 0 no southeast 1629.833 female no
## 1337 21 female 25.800 0 no southwest 2007.945 female no
## 1338 61 female 29.070 0 yes northwest 29141.360 female yes
## f.region
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## 1338 northwest#There is only one observation which repeat twice, it makes sense that a person with the same properties will have the same charge and since it's only one we decide to leave it there.
#outliersOutlier detection
Univariate
We can see extreme outliers for both charges and bmi, since it’s just serval observation it might be the case that for a certain bmi, age or smokers the charge value is raising by a lot compare to the rest. from looking at the high value of column charges it can be seen that all are smokers and mid-high bmi, also some of the ages I see are relatively high. For the target variable we can see there is no lower bound for extreme and mild outliers, it’s also can be seen on the Boxplot(). For variable bmi, mild outliers on the upper bound and no sever upper bound outliers and not lower bound outliers. We decided to delete the 6 univariate outliers since the charges are very high, even though all 6 observation are smokers, there are 274 smokers in the dataset and their charges values are not as high as the extreme outliers observations
par(mfrow=c(1,2))
Boxplot(df$charges)## [1] 544 1301 1231 578 820 1147 35 1242 1063 489Boxplot(df$bmi)
## [1] 117 287 402 544 848 861 1048 1089 1318Boxplot(df$age)
Boxplot(df$children)
#
# treat outliers for charges variable
sevout<-quantile(df$charges,0.75,na.rm=TRUE)+3*(quantile(df$charges,0.75,na.rm=TRUE)-quantile(df$charges,0.25,na.rm=TRUE))
sevout## 75%
## 52338.79sev_out_lower <- quantile(df$charges,0.25,na.rm=TRUE)-3*(quantile(df$charges,0.75,na.rm=TRUE)-quantile(df$charges,0.25,na.rm=TRUE))
mist<-quantile(df$charges,0.75,na.rm=TRUE)+1.5*(quantile(df$charges,0.75,na.rm=TRUE)-quantile(df$charges,0.25,na.rm=TRUE))
mist## 75%
## 34489.35mist_out_lower <- quantile(df$charges,0.25,na.rm=TRUE)-1.5*(quantile(df$charges,0.75,na.rm=TRUE)-quantile(df$charges,0.25,na.rm=TRUE))
# get list of outliers
loutse<-which(df$charges>sevout);length(loutse)## [1] 6loutmist <-which(df$charges>mist);length(loutmist)## [1] 139low_out_sever <- which(df$charges<sev_out_lower);low_out_sever## integer(0)low_out_mild <- which(df$charges<mist_out_lower);low_out_mild## integer(0)table(loutse)## loutse
## 544 578 820 1147 1231 1301
## 1 1 1 1 1 1table(loutmist)## loutmist
## 15 20 24 30 31 35 39 40 50 54 56 83 85 87 95 110
## 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## 124 147 159 162 176 186 204 224 241 243 252 253 255 257 264 266
## 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## 272 282 289 293 299 313 315 323 328 329 331 339 374 378 382 421
## 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## 422 423 442 477 489 501 525 531 544 550 559 570 578 588 610 616
## 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## 622 624 630 666 668 669 675 678 683 690 698 707 726 737 739 740
## 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## 743 760 804 820 827 829 843 846 851 853 857 861 884 894 902 918
## 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## 948 952 954 957 959 1013 1022 1023 1032 1037 1038 1048 1050 1063 1071 1079
## 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## 1091 1097 1112 1118 1119 1123 1125 1140 1147 1153 1157 1187 1207 1208 1219 1231
## 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## 1241 1242 1250 1285 1289 1292 1301 1302 1304 1314 1324
## 1 1 1 1 1 1 1 1 1 1 1# see outliers
Boxplot(df$charges)## [1] 544 1301 1231 578 820 1147 35 1242 1063 489abline(h=sevout,col="red")
abline(h=mist,col="yellow")
# Since there are only 6 severe outliers, we will remove them from the dataset,
df <- df[-which(df$charges >= sevout),]
# check severe outliers for bmi atrribute
sevout_bmi<-quantile(df$bmi,0.75,na.rm=TRUE)+3*(quantile(df$bmi,0.75,na.rm=TRUE)-quantile(df$bmi,0.25,na.rm=TRUE));sevout_bmi## 75%
## 59.815mist_bmi <- quantile(df$bmi,0.75,na.rm=TRUE)+1.5*(quantile(df$bmi,0.75,na.rm=TRUE)-quantile(df$bmi,0.25,na.rm=TRUE))
loutse_bmi<-which(df$bmi>sevout_bmi);length(loutse_bmi) # no severe outliers for bmi## [1] 0colSums(is.na(df))## age sex bmi children smoker region charges f.sex
## 0 0 0 0 0 0 0 0
## f.smoker f.region
## 0 0serout_lower_bmi <- quantile(df$bmi,0.25,na.rm=TRUE)-3*(quantile(df$bmi,0.75,na.rm=TRUE)-quantile(df$bmi,0.25,na.rm=TRUE));serout_lower_bmi## 25%
## 1.02375mist_lower_bmi <- quantile(df$bmi,0.25,na.rm=TRUE)-1.5*(quantile(df$bmi,0.75,na.rm=TRUE)-quantile(df$bmi,0.25,na.rm=TRUE));mist_lower_bmi## 25%
## 13.62187up_sever_bmi <- which(df$bmi > sevout_bmi); up_sever_bmi## integer(0)up_mild_bmi <- which(df$bmi > mist_bmi); up_mild_bmi## [1] 117 287 402 845 858 1045 1086 1312low_sever_bmi <- which(df$bmi < serout_lower_bmi); low_sever_bmi## integer(0)low_mild_bmi <- which(df$bmi < mist_lower_bmi); low_mild_bmi## integer(0)
Multivariate
For the multivariate outliers, we have chosen the quantile to be a very high value so outliers we get are very extreme compare to our values in the dataset. Observation number 1048 is the multivatiate outlier we have got and it’s indeed a very high value of charge and bmi. Since this observation is so extreme we will remove it from the dataset. We see from the plot classical Mahalanobis distance vs robust Mahalanobis distance that there is one observation (1048) that is behind the cutoff value, in addition we can indicate 3 clusters and number of observations that a bit far from the clusters, it can be suspected as influential data. We also plot charges vs bmi and we can see on the top right corner of the graph there is one observation which has high charge and bmi.
res.out<-Moutlier(df[,c(7,3,1,4)],quantile=0.999)
#str(res.out)
plot(df$charges,df$bmi)
res.out$cutoff## [1] 4.297305#quantile(res.out$md,seq(0,1,0.001))
which((res.out$md > res.out$cutoff) & (res.out$rd > res.out$cutoff))## 1048
## 1045plot( res.out$md, res.out$rd )
#text(res.out$md, res.out$rd, labels=rownames(df),adj=1, cex=0.5)
abline(h=res.out$cutoff, col="red")
abline(v=res.out$cutoff, col="red")
df <- df[-which(res.out$md > res.out$cutoff & res.out$rd > res.out$cutoff),]
#res.out$cutoff^2
#qchisq(0.975,4)
#aq.plot(df[,c(3,7)],delta = qchisq(0.95,df=ncol(x)),alpha = 0.05)
#THIS 3 LINES I THINK WE CAN DELETE- Detect univariant and multivariant outliers, errors and missing values (if any) and apply animputation technique if needed. [Achraf]
(taking into acount all features) ###I THINK WE CAN DELETE THE WHILE CHUNK BELO
Missing data
# check missing data
# Imputating using median is used in the numeric variable "charges" for severe outliers
# there is no missing data in the dataframe so no further imputation is needed
colSums(is.na(df))## age sex bmi children smoker region charges f.sex
## 0 0 0 0 0 0 0 0
## f.smoker f.region
## 0 0Data Validation
After doing the pre processing steps where we detected and removed outliers, we will check if data makes sense using common sense and domain knowledge.
summary(df)## age sex bmi children
## Min. :18.00 Length:1331 Min. :15.96 Min. :0.000
## 1st Qu.:26.50 Class :character 1st Qu.:26.22 1st Qu.:0.000
## Median :39.00 Mode :character Median :30.30 Median :1.000
## Mean :39.19 Mean :30.62 Mean :1.097
## 3rd Qu.:51.00 3rd Qu.:34.60 3rd Qu.:2.000
## Max. :64.00 Max. :53.13 Max. :5.000
## smoker region charges f.sex f.smoker
## Length:1331 Length:1331 Min. : 1122 female:659 no :1064
## Class :character Class :character 1st Qu.: 4720 male :672 yes: 267
## Mode :character Mode :character Median : 9302
## Mean :13042
## 3rd Qu.:16359
## Max. :51195
## f.region
## northeast:323
## northwest:323
## southeast:361
## southwest:324
##
## We have ages ranging from 18 to 64, and which bmi ranging from 16 to 53 which are values that are in the following table. The balance between factor variable si really good. However, only 20% of the sample are smokers.
Let’s see how the relationship between children per age.
plot(df$children~df$age)
As we can see in the plot, there are individuals with age 20 that have from 3 to 5 children which is really strange.
thr2five_children <- which(df$age <= 20 & df$children>2)
thr2five_children## [1] 33 167 370 982 1092 1182 1191 1200This observations will be removed since it’s something very unlikely.
df <- df[-thr2five_children,]Let’s check now the bmi values per age to see if there is any weird case:
https://patient.info/doctor/bmi-calculator-calculator
plot(df$bmi~df$age)
In this case the plot shows there are young people who have a really high bmi. Since data is from EEUU, and there are a lof of obesity problems, we decide that these observations are not going to be removed.
summary(df)## age sex bmi children
## Min. :18.00 Length:1323 Min. :15.96 Min. :0.00
## 1st Qu.:27.00 Class :character 1st Qu.:26.22 1st Qu.:0.00
## Median :39.00 Mode :character Median :30.30 Median :1.00
## Mean :39.31 Mean :30.62 Mean :1.08
## 3rd Qu.:51.00 3rd Qu.:34.60 3rd Qu.:2.00
## Max. :64.00 Max. :53.13 Max. :5.00
## smoker region charges f.sex f.smoker
## Length:1323 Length:1323 Min. : 1122 female:654 no :1058
## Class :character Class :character 1st Qu.: 4729 male :669 yes: 265
## Mode :character Mode :character Median : 9305
## Mean :13047
## 3rd Qu.:16265
## Max. :51195
## f.region
## northeast:320
## northwest:321
## southeast:360
## southwest:322
##
## Explanatory data analysis
summary(df)## age sex bmi children
## Min. :18.00 Length:1323 Min. :15.96 Min. :0.00
## 1st Qu.:27.00 Class :character 1st Qu.:26.22 1st Qu.:0.00
## Median :39.00 Mode :character Median :30.30 Median :1.00
## Mean :39.31 Mean :30.62 Mean :1.08
## 3rd Qu.:51.00 3rd Qu.:34.60 3rd Qu.:2.00
## Max. :64.00 Max. :53.13 Max. :5.00
## smoker region charges f.sex f.smoker
## Length:1323 Length:1323 Min. : 1122 female:654 no :1058
## Class :character Class :character 1st Qu.: 4729 male :669 yes: 265
## Mode :character Mode :character Median : 9305
## Mean :13047
## 3rd Qu.:16265
## Max. :51195
## f.region
## northeast:320
## northwest:321
## southeast:360
## southwest:322
##
## #numeric variables
summary(df[,c(1,3,4,7)]) ## age bmi children charges
## Min. :18.00 Min. :15.96 Min. :0.00 Min. : 1122
## 1st Qu.:27.00 1st Qu.:26.22 1st Qu.:0.00 1st Qu.: 4729
## Median :39.00 Median :30.30 Median :1.00 Median : 9305
## Mean :39.31 Mean :30.62 Mean :1.08 Mean :13047
## 3rd Qu.:51.00 3rd Qu.:34.60 3rd Qu.:2.00 3rd Qu.:16265
## Max. :64.00 Max. :53.13 Max. :5.00 Max. :51195#plot(df[,c(1,3,4,7)])
ggpairs(df[,c(1,3,4,7)])
#categorical variables
summary(df[,c(1,4,8:10)])## age children f.sex f.smoker f.region
## Min. :18.00 Min. :0.00 female:654 no :1058 northeast:320
## 1st Qu.:27.00 1st Qu.:0.00 male :669 yes: 265 northwest:321
## Median :39.00 Median :1.00 southeast:360
## Mean :39.31 Mean :1.08 southwest:322
## 3rd Qu.:51.00 3rd Qu.:2.00
## Max. :64.00 Max. :5.00From the summary we can see the factor values, it seems that sex and region are distributed equally and not much smokers compare to the non smokers. age and number of children looks about right and there is values in a range that makes sense. In addition, we see low correlation (0.198) between the target variable and the other numeric explantory variable bmi. We don’t see any pattern in the relation between the two variables. We see number of extreme values with high bmi and/or charges.
- Determine if the response variable (charges) has an acceptably normal distribution.
# Density plot to check the distribution
ggpubr::ggdensity(df$charges, fill = "lightgray", add = "mean", xlab = "charges variable density")## Warning: `geom_vline()`: Ignoring `mapping` because `xintercept` was provided.## Warning: `geom_vline()`: Ignoring `data` because `xintercept` was provided.## Warning: The dot-dot notation (`..density..`) was deprecated in ggplot2 3.4.0.
## ℹ Please use `after_stat(density)` instead.
## ℹ The deprecated feature was likely used in the ggpubr package.
## Please report the issue at <]8;;https://github.com/kassambara/ggpubr/issueshttps://github.com/kassambara/ggpubr/issues]8;;>.
# Shapiro Test to asses that data on response variable is normaly distribution
# H0 = Data is normally distributed
# H1 = Data is not normally distributed
# alfa = 0.05
shapiro.test(df$charges)##
## Shapiro-Wilk normality test
##
## data: df$charges
## W = 0.81754, p-value < 2.2e-16As we can see, the density plot shows that data is not normally distributed. To asses that, we can use one of many statistical tests that check normality on data. In this case, we use Shapiro test.
The result of the Shapiro test shows that data in variable charges is not normally distributed since p-value is less than the significance level (0.05) so we reject the null hypothesis (data is normally distributed) and we conclude that data is not normally distributed (alternative hypothesis)
Let’s try to apply the log transformation
# Density plot to check the distribution
ggpubr::ggdensity(log(df$charges), fill = "lightgray", add = "mean", xlab = "charges variable density")## Warning: `geom_vline()`: Ignoring `mapping` because `xintercept` was provided.## Warning: `geom_vline()`: Ignoring `data` because `xintercept` was provided.
# Shapiro Test to asses that data on response variable is normaly distribution
# H0 = Data is normally distributed
# H1 = Data is not normally distributed
# alfa = 0.05
shapiro.test(log(df$charges))##
## Shapiro-Wilk normality test
##
## data: log(df$charges)
## W = 0.98152, p-value = 5.679e-12The null hypothesis can be still rejected so data still not being not normally distributed.
par(mfrow=c(1,1))
acf(df$charges)
dwtest(df$charges~1)##
## Durbin-Watson test
##
## data: df$charges ~ 1
## DW = 2.0054, p-value = 0.5394
## alternative hypothesis: true autocorrelation is greater than 0Address tests to discard serial correlation: In the acf (auto correlation function) we can see from the graph that the data is not correlated where we have the blue threshold and all lines are within the threshold, we do see that there is one or two lines that crosses the threshold but just in a little bit so we leave it as it is without random the order of the observations. In addition we address Durbin-Watson test to check whether true autocorrelation is greater or not than 0. We see p-value 0.5183, thus we don’t reject the null hypothesis and say that true autocorrelation is not greater than 0.
- Preliminary exploratory analysis to describe the relationships observed has to be undertaken. [Eliya]
Association to the target variable, we see the numeric variable age 0.301 which is the most associated but the number is quite low and it is not strong association. f.smoker is globally associated to charges, in particular, f.smoker=yes is very remarkable.
#library(DataExplorer)
#create_report(df, y= "charges")
library(FactoMineR)
res.con <- condes(df[,c(1,3,4,7,8:10)], num.var = 4 , proba = 0.01 )
res.con$quanti## correlation p.value
## age 0.30679657 3.128392e-30
## bmi 0.18280602 2.091908e-11
## children 0.08239851 2.705520e-03res.con$quali## R2 p.value
## f.smoker 0.6169962 1.418037e-277res.con$category## Estimate p.value
## f.smoker=yes 11493.36 1.418037e-277
## f.smoker=no -11493.36 1.418037e-277#THIS IS NOTE TO OURSELFS - there this libray which basically creates a whole report of the explenatory data analysis, we can consider if we want to put into as EDA is requested twice in the project statement, once at data preparation and another on the tasks [Achraf: For me OK!]
- Apart from the original factor variables, you can consider other categorical variables that can be defined from categorized numeric variables. [Eliya]
We have created a new variable called age_range where we divide the ages into 4 groups according to the 4 quantiles. From the summary (and the new column in the data set) we see 4 groups of ages and how many observations were fit into each age group.
df$age_range <- cut(df$age, breaks = quantile(df$age,probs = c(0,0.25,0.5,0.75,1)), include.lowest = T)
summary(df)## age sex bmi children
## Min. :18.00 Length:1323 Min. :15.96 Min. :0.00
## 1st Qu.:27.00 Class :character 1st Qu.:26.22 1st Qu.:0.00
## Median :39.00 Mode :character Median :30.30 Median :1.00
## Mean :39.31 Mean :30.62 Mean :1.08
## 3rd Qu.:51.00 3rd Qu.:34.60 3rd Qu.:2.00
## Max. :64.00 Max. :53.13 Max. :5.00
## smoker region charges f.sex f.smoker
## Length:1323 Length:1323 Min. : 1122 female:654 no :1058
## Class :character Class :character 1st Qu.: 4729 male :669 yes: 265
## Mode :character Mode :character Median : 9305
## Mean :13047
## 3rd Qu.:16265
## Max. :51195
## f.region age_range
## northeast:320 [18,27]:353
## northwest:321 (27,39]:310
## southeast:360 (39,51]:336
## southwest:322 (51,64]:324
##
## Building the model
If you can improve linear relations or limit the effect of influential data, you must consider the suitable transformations for variables. [Achraf]
When building the model, you should study the presence of multicollinearity and try to reduce their impact on the model for easier interpretation. [Achraf]
You should build the model using a technique for selecting variables (removing no significant predictors and/or stepwise selection of the best models). [Achraf]
The validation of the model has to be done with graphs and / or suitable tests to verify model assumptions. [Achraf]
You must include the study of unusual and / or influential data. [Achraf]
First model
par(mfrow=c(1,1))
plot(df$charges,df$bmi,pch=19)
#text(df$charges,df$bmi,label=row.names(df),col="darkgreen",adj=1.5)
m1<-lm(charges~bmi+age+children, data = df)
summary(m1)##
## Call:
## lm(formula = charges ~ bmi + age + children, data = df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -12628 -6735 -5057 5894 39232
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -5853.05 1710.99 -3.421 0.000643 ***
## bmi 288.72 50.14 5.758 1.06e-08 ***
## age 239.14 21.79 10.977 < 2e-16 ***
## children 610.04 255.28 2.390 0.017003 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 11000 on 1319 degrees of freedom
## Multiple R-squared: 0.1202, Adjusted R-squared: 0.1182
## F-statistic: 60.05 on 3 and 1319 DF, p-value: < 2.2e-16lines(df$bmi,fitted(m1),col="red")
par(mfrow=c(2,2))
plot(m1)
par(mfrow=c(1,1))Looking at the summary of the model, the RSquared is very low and there is a lot of residual standard error.
If we study the residual error looking at the plots we can see that the data is not following a normal distribution since there are deviations of the line (Normal Q-Q plot). Also there are a lot of sparsity in the variance (Scale-Location plot).
Asses multicollinearity
Maybe there is multicollinearity that is causing bad results
car::vif(m1)## bmi age children
## 1.012957 1.017013 1.004239The vif values are low (less than 5) so there aren’t problems of multidisciplinary.
Let’s try to do some transformations to the data.
Transformation
library(MASS)
boxcox(charges~bmi+age+children, data = df)
The boxplots shows that the lambda values are close to 0 so a logarithmic transformation to the target variable should help to improve the results
# (only for numerical variables)
boxTidwell(log(charges) ~ bmi + age + I(children+0.5), data=df)## MLE of lambda Score Statistic (z) Pr(>|z|)
## bmi -1.07828 -1.4110 0.15824
## age 0.42692 -1.7687 0.07694 .
## I(children + 0.5) 0.25004 -1.7969 0.07235 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## iterations = 16# poly(age,3) for adding ortogonal polynomials- TODO: check transformations for explanatory variables
par(mfrow=c(1,1))
# apply logarithm to the charges variable
plot(log(df$charges),df$bmi,pch=19)
m2 <- lm(log(charges)~log(bmi)+ age+children, data = df)
lines(df$bmi, fitted(m2), color="red")## Warning in plot.xy(xy.coords(x, y), type = type, ...): "color" is not a
## graphical parameter
summary(m2)##
## Call:
## lm(formula = log(charges) ~ log(bmi) + age + children, data = df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.3597 -0.4333 -0.3057 0.4823 2.2139
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 6.652686 0.352005 18.899 < 2e-16 ***
## log(bmi) 0.291298 0.103857 2.805 0.00511 **
## age 0.033831 0.001503 22.511 < 2e-16 ***
## children 0.107245 0.017595 6.095 1.43e-09 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.7584 on 1319 degrees of freedom
## Multiple R-squared: 0.311, Adjusted R-squared: 0.3094
## F-statistic: 198.4 on 3 and 1319 DF, p-value: < 2.2e-16par(mfrow=c(2,2))
plot(m2)
par(mfrow=c(1,1))The model is still not performing very well. However if we check the study of residuals we can see that it results in an improvement.
The normal Q-Q plot still have a deviation but is that big as the m1 and if we check the Scale-Location of the standard residuals the variance is better.
avPlots(m2)
The partial regressions plots shows that all regresors have two big clusters of data.
AIC(m1,m2)## df AIC
## m1 5 28383.903
## m2 5 3028.666The AIC test shows that model 2 is performing much better than model 1 so we will continue with it.
Inlfuential data
Maybe, removing influential data the results can be improved.
Residual outliers
Influential values
library(car)
influencePlot(m2)
## StudRes Hat CookD
## 439 -0.8322950 0.012634066 0.002216465
## 804 2.9361024 0.005707346 0.012299867
## 1086 0.6614852 0.013676343 0.001517456
## 1140 2.9223204 0.003101127 0.006603720
## 1157 2.9000979 0.006456079 0.013586697# there are a lot of influential data
# Manually removing influential points
ll <- which(rownames(df) %in% c("1048", "848", "1318", "443")); ll## [1] 440 842 1303m3 <- lm(log(charges)~log(bmi)+age+children, data=df[-ll,])
summary(m3)##
## Call:
## lm(formula = log(charges) ~ log(bmi) + age + children, data = df[-ll,
## ])
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.3222 -0.4349 -0.3062 0.4904 2.1926
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 6.550608 0.353456 18.533 < 2e-16 ***
## log(bmi) 0.326430 0.104491 3.124 0.00182 **
## age 0.033497 0.001505 22.258 < 2e-16 ***
## children 0.105806 0.017573 6.021 2.25e-09 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.757 on 1316 degrees of freedom
## Multiple R-squared: 0.3087, Adjusted R-squared: 0.3071
## F-statistic: 195.9 on 3 and 1316 DF, p-value: < 2.2e-16par(mfrow=c(2,2))
plot(m3)
par(mfrow=c(1,1))
influencePlot(m3)
## StudRes Hat CookD
## 439 -0.8450308 0.012659570 0.002289452
## 804 2.9102984 0.005833903 0.012355409
## 1086 0.6895374 0.013769613 0.001660245
## 1140 2.9091690 0.003144767 0.006637099
## 1157 2.8718487 0.006603252 0.013630539# With cooks distance
cooksD <- cooks.distance(m2)
n <- nrow(df)
plot(cooksD, main = "Cooks Distance for Influential Obs")
abline(h = 4/n, lty = 2, col = "steelblue") # add cutoff line
# TODO: GET A BETTER THRESHOLD ()
influential_obs <- as.numeric(names(cooksD)[(cooksD > (4/n))])
influential_obs## [1] 15 20 31 35 58 65 83 103 129 158 159 162 186 204 220
## [16] 224 241 251 260 264 293 299 315 322 363 378 413 431 443 477
## [31] 495 501 504 517 527 550 610 619 622 624 726 737 739 740 760
## [46] 782 804 843 861 912 990 1002 1020 1022 1028 1034 1037 1040 1043 1094
## [61] 1118 1121 1125 1140 1157 1197 1224 1232 1268 1283 1289 1292 1309 1314 1318length(influential_obs)## [1] 75m4 <- lm(log(charges)~log(bmi)+age+children, data=df[-influential_obs,])
summary(m4)##
## Call:
## lm(formula = log(charges) ~ log(bmi) + age + children, data = df[-influential_obs,
## ])
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.3447 -0.4259 -0.3026 0.4622 2.2308
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 6.597621 0.358669 18.395 < 2e-16 ***
## log(bmi) 0.299185 0.105587 2.834 0.00468 **
## age 0.034312 0.001527 22.466 < 2e-16 ***
## children 0.109236 0.017969 6.079 1.6e-09 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.7492 on 1244 degrees of freedom
## Multiple R-squared: 0.3213, Adjusted R-squared: 0.3196
## F-statistic: 196.3 on 3 and 1244 DF, p-value: < 2.2e-16par(mfrow=c(2,2))
plot(m4)
par(mfrow=c(1,1))
influencePlot(m4)
## StudRes Hat CookD
## 439 -0.8566140 0.013481281 0.002507432
## 804 2.9962837 0.006021808 0.013510757
## 1086 0.6745291 0.014553091 0.001680559
## 1140 2.9842597 0.003301141 0.007327605
## 1157 2.9585630 0.006808004 0.014906989#create scatterplot with outliers present
outliers_present <- ggplot(data = df, aes(x = log(bmi) + age + children, y = log(charges))) +
geom_point() +
geom_smooth(method = lm) +
# ylim(0, 200) +
ggtitle("Ifluential data Present")
#create scatterplot with outliers removed
outliers_removed <- ggplot(data = df[-influential_obs,], aes(x = log(bmi) + age + children, y = log(charges))) +
geom_point() +
geom_smooth(method = lm) +
# ylim(0, 200) +
ggtitle("Influential data Removed")
#plot both scatterplots side by side
gridExtra::grid.arrange(outliers_present, outliers_removed, ncol = 2)## `geom_smooth()` using formula = 'y ~ x'
## `geom_smooth()` using formula = 'y ~ x'
Adding factors (very important)
Check that meaning of a factor could not be related to the numerical variables so one should be used.
AIC test to compare
summary(df)## age sex bmi children
## Min. :18.00 Length:1323 Min. :15.96 Min. :0.00
## 1st Qu.:27.00 Class :character 1st Qu.:26.22 1st Qu.:0.00
## Median :39.00 Mode :character Median :30.30 Median :1.00
## Mean :39.31 Mean :30.62 Mean :1.08
## 3rd Qu.:51.00 3rd Qu.:34.60 3rd Qu.:2.00
## Max. :64.00 Max. :53.13 Max. :5.00
## smoker region charges f.sex f.smoker
## Length:1323 Length:1323 Min. : 1122 female:654 no :1058
## Class :character Class :character 1st Qu.: 4729 male :669 yes: 265
## Mode :character Mode :character Median : 9305
## Mean :13047
## 3rd Qu.:16265
## Max. :51195
## f.region age_range
## northeast:320 [18,27]:353
## northwest:321 (27,39]:310
## southeast:360 (39,51]:336
## southwest:322 (51,64]:324
##
## m5 <- lm(log(charges)~log(bmi)+age+children+f.sex+f.smoker+f.region+age_range, data=df[-influential_obs,])summary(m5)##
## Call:
## lm(formula = log(charges) ~ log(bmi) + age + children + f.sex +
## f.smoker + f.region + age_range, data = df[-influential_obs,
## ])
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.98041 -0.19855 -0.06082 0.05858 2.18942
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 6.074598 0.230028 26.408 < 2e-16 ***
## log(bmi) 0.397670 0.064550 6.161 9.78e-10 ***
## age 0.033779 0.003629 9.308 < 2e-16 ***
## children 0.096003 0.011021 8.711 < 2e-16 ***
## f.sexmale -0.076837 0.024980 -3.076 0.002144 **
## f.smokeryes 1.528461 0.031532 48.473 < 2e-16 ***
## f.regionnorthwest -0.062135 0.035608 -1.745 0.081237 .
## f.regionsoutheast -0.146053 0.035864 -4.072 4.95e-05 ***
## f.regionsouthwest -0.125665 0.035982 -3.492 0.000496 ***
## age_range(27,39] 0.083803 0.056098 1.494 0.135464
## age_range(39,51] 0.060289 0.093743 0.643 0.520260
## age_range(51,64] 0.053163 0.135782 0.392 0.695473
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4394 on 1236 degrees of freedom
## Multiple R-squared: 0.768, Adjusted R-squared: 0.7659
## F-statistic: 372 on 11 and 1236 DF, p-value: < 2.2e-16Anova(m5)## Anova Table (Type II tests)
##
## Response: log(charges)
## Sum Sq Df F value Pr(>F)
## log(bmi) 7.33 1 37.9539 9.783e-10 ***
## age 16.73 1 86.6436 < 2.2e-16 ***
## children 14.65 1 75.8850 < 2.2e-16 ***
## f.sex 1.83 1 9.4614 0.0021445 **
## f.smoker 453.67 1 2349.6590 < 2.2e-16 ***
## f.region 3.88 3 6.6945 0.0001755 ***
## age_range 0.83 3 1.4327 0.2315802
## Residuals 238.64 1236
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1#remove age range
m6 <- lm(log(charges)~log(bmi)+age+children+f.sex+f.smoker+f.region, data=df[-influential_obs,])
anova(m6, m5)## Analysis of Variance Table
##
## Model 1: log(charges) ~ log(bmi) + age + children + f.sex + f.smoker +
## f.region
## Model 2: log(charges) ~ log(bmi) + age + children + f.sex + f.smoker +
## f.region + age_range
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 1239 239.47
## 2 1236 238.64 3 0.82986 1.4327 0.2316# models are not equivalent.
AIC(m6, m5)## df AIC
## m6 10 1501.407
## m5 13 1503.074m7 <- step( m5 )## Start: AIC=-2040.6
## log(charges) ~ log(bmi) + age + children + f.sex + f.smoker +
## f.region + age_range
##
## Df Sum of Sq RSS AIC
## - age_range 3 0.83 239.47 -2042.3
## <none> 238.64 -2040.6
## - f.sex 1 1.83 240.47 -2033.1
## - f.region 3 3.88 242.52 -2026.5
## - log(bmi) 1 7.33 245.97 -2004.8
## - children 1 14.65 253.30 -1968.2
## - age 1 16.73 255.37 -1958.0
## - f.smoker 1 453.67 692.31 -713.4
##
## Step: AIC=-2042.26
## log(charges) ~ log(bmi) + age + children + f.sex + f.smoker +
## f.region
##
## Df Sum of Sq RSS AIC
## <none> 239.47 -2042.26
## - f.sex 1 1.84 241.31 -2034.72
## - f.region 3 3.85 243.32 -2028.37
## - log(bmi) 1 7.25 246.72 -2007.07
## - children 1 18.01 257.48 -1953.78
## - age 1 291.71 531.19 -1050.02
## - f.smoker 1 454.48 693.95 -716.44summary(m7)##
## Call:
## lm(formula = log(charges) ~ log(bmi) + age + children + f.sex +
## f.smoker + f.region, data = df[-influential_obs, ])
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.99813 -0.19942 -0.05369 0.06068 2.16261
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 6.0817177 0.2171276 28.010 < 2e-16 ***
## log(bmi) 0.3952837 0.0645613 6.123 1.23e-09 ***
## age 0.0348632 0.0008974 38.849 < 2e-16 ***
## children 0.1019526 0.0105624 9.652 < 2e-16 ***
## f.sexmale -0.0770675 0.0249914 -3.084 0.002089 **
## f.smokeryes 1.5289068 0.0315295 48.491 < 2e-16 ***
## f.regionnorthwest -0.0627494 0.0356199 -1.762 0.078377 .
## f.regionsoutheast -0.1452666 0.0358791 -4.049 5.47e-05 ***
## f.regionsouthwest -0.1259063 0.0359944 -3.498 0.000485 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4396 on 1239 degrees of freedom
## Multiple R-squared: 0.7672, Adjusted R-squared: 0.7657
## F-statistic: 510.4 on 8 and 1239 DF, p-value: < 2.2e-16par( mfrow = c(2,2))
plot( m7, id.n=0 )
par( mfrow = c(1,1))- Redefining factors
# New categorical variables can be extracted from the actual ones?
?boxTidwell
boxTidwell(log(charges) ~ bmi + age + I(children+0.5),~f.sex+f.smoker+f.region, data=df[-influential_obs,])## MLE of lambda Score Statistic (z) Pr(>|z|)
## bmi -1.19511 -2.8929 0.003817 **
## age 0.51132 -2.8293 0.004665 **
## I(children + 0.5) 0.43209 -1.5489 0.121396
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## iterations = 11Adding interactions (very imporant)
Interactions between all factors (not a problem)
Double interactions (ONLY)
Factor x factor
Factor x numerical
plot AllEffects of partial regression to check what the model is doing with the interactions
m8 <- lm(log(charges)~log(bmi)+age
+children * (f.sex+f.smoker+f.region+age_range), data=df[-influential_obs,])
summary(m8)##
## Call:
## lm(formula = log(charges) ~ log(bmi) + age + children * (f.sex +
## f.smoker + f.region + age_range), data = df[-influential_obs,
## ])
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.96864 -0.21409 -0.06253 0.05899 2.28432
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 6.00831 0.22709 26.458 < 2e-16 ***
## log(bmi) 0.41086 0.06325 6.496 1.20e-10 ***
## age 0.03020 0.00361 8.366 < 2e-16 ***
## children 0.23773 0.03237 7.344 3.75e-13 ***
## f.sexmale -0.09270 0.03309 -2.802 0.005162 **
## f.smokeryes 1.68238 0.04275 39.350 < 2e-16 ***
## f.regionnorthwest -0.07351 0.04764 -1.543 0.123072
## f.regionsoutheast -0.17956 0.04629 -3.879 0.000110 ***
## f.regionsouthwest -0.09628 0.04718 -2.041 0.041484 *
## age_range(27,39] 0.23394 0.06471 3.615 0.000312 ***
## age_range(39,51] 0.29659 0.10079 2.943 0.003313 **
## age_range(51,64] 0.30292 0.14105 2.148 0.031935 *
## children:f.sexmale 0.01498 0.02086 0.718 0.472951
## children:f.smokeryes -0.13681 0.02666 -5.131 3.35e-07 ***
## children:f.regionnorthwest 0.01362 0.02993 0.455 0.649076
## children:f.regionsoutheast 0.02853 0.02916 0.978 0.328058
## children:f.regionsouthwest -0.02515 0.02932 -0.858 0.391264
## children:age_range(27,39] -0.14410 0.03129 -4.605 4.55e-06 ***
## children:age_range(39,51] -0.17813 0.03159 -5.640 2.11e-08 ***
## children:age_range(51,64] -0.16985 0.03243 -5.237 1.92e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4294 on 1228 degrees of freedom
## Multiple R-squared: 0.7799, Adjusted R-squared: 0.7765
## F-statistic: 229.1 on 19 and 1228 DF, p-value: < 2.2e-16par( mfrow = c(2,2))
plot( m8, id.n=0 )
par( mfrow = c(1,1))
#?lm
avPlots(m8)
#library(effects)
#plot(allEffects(m8))Validation of the model
Address residual outliers, influential data
It should be robust for exploratory variables (iterations)
Check if the models could be improved in some way
We should not expect a great coefficient of determination at the end (LIDIA)
- Maybe: Some critical explanatory variable that is missing
library(effects)## lattice theme set by effectsTheme()
## See ?effectsTheme for details.?allEffects